On Noncentral Tanny-Dowling Polynomials and Generalizations of Some Formulas for Geometric Polynomials
aa r X i v : . [ m a t h . G M ] A p r On Noncentral Tanny-Dowling Polynomialsand Generalizations of Some Formulasfor Geometric Polynomials
Mahid M. Mangontarum and Norlailah M. Madid Department of MathematicsMindanao State University–Main CampusMarawi City 9700Philippines [email protected] [email protected] [email protected] Abstract
In this paper, we establish some formulas for the noncentral Tanny-Dowling poly-nomials including sums of products and explicit formulas which are shown to be gen-eralizations of known identities. Other important results and consequences are alsodiscussed and presented.
The geometric polynomials [16], denoted by w n ( x ), are defined by w n ( x ) = n X k =0 k ! (cid:26) nk (cid:27) x k , (1)where (cid:8) nk (cid:9) are the well-celebrated Stirling numbers of the second kind [6, 15]. These poly-nomials are known to satisfy the exponential generating function ∞ X n =0 w n ( x ) z n n ! = 11 − x ( e z −
1) (2)and the recurrence relation w n +1 ( x ) = x ddx [ w n ( x ) + xw n ( x )] . (3)The case when x = 1 yields w n := w n (1) = n X k =0 k ! (cid:26) nk (cid:27) , (4)1he geometric numbers (or ordered Bell numbers). Recall that the numbers (cid:8) nk (cid:9) countthe number of partitions of a set X with n elements into k non-empty subsets. Thesenumbers can also be interpreted as the number of ways to distribute n distinct objects into k identical boxes such that no box is empty. On the other hand, the numbers k ! (cid:8) nk (cid:9) can becombinatorially interpreted as the number of distinct ordered partitions of X with k blocks,or the numbers of ways to distribute n distinct objects into k distinct boxes. It followsimmediately that the geometric numbers count the number of distinct ordered partitions ofthe n -set X .The study of geometric polynomials and numbers has a long history. Aside from thepaper of Tanny [16], one may also see the works of Boyadzhiev [4], Dil and Kurt [8], and thereferences therein for further readings. Benoumhani [3] studied two equivalent generalizationsof w n ( x ) given by F m, ( n ; x ) = n X k =0 m k k ! W m ( n, k ) x k (5)and F m, ( n ; x ) = n X k =0 k ! W m ( n, k ) x k , (6)where W m ( n, k ) denote the Whitney numbers of the second kind of Dowling lattices [2].These are called Tanny-Dowling polynomials and are known to satisfy the following expo-nential generating functions: ∞ X n =0 F m, ( x ) z n n ! = e z − x ( e mz − , (7) ∞ X n =0 F m, ( x ) z n n ! = e z − xm ( e mz − . (8)More properties can be seen in [2, 3]. In a recent paper, Kargın [9] established a number ofexplicit formulas and formulas involving products of geometric polynomials, viz.( x + 1) n X k =0 (cid:18) nk (cid:19) w k ( x ) w n − k ( x ) = w n +1 ( x ) + w n ( x ) , (9) n X k =0 (cid:18) nk (cid:19) w k ( x ) w n − k ( x ) = x w n ( x ) − x w n ( x ) x − x , (10) w n ( x ) = x n X k =1 (cid:26) nk (cid:27) ( − n + k k !( x + 1) k − , (11)and w n ( x ) = n X k =0 (cid:26) nk (cid:27) k ! x k n +1 ( x + 1) x k + ( − k +1 (2 x + 1) k +1 . (12)2his was done with the aid of the two-variable geometric polynomials w k ( r ; x ) defined by ∞ X n =0 w n ( r ; x ) z n n ! = e rz − x ( e z − . (13)A natural generalization of F m, ( x ) and F m, ( x ) are the noncentral Tanny-Dolwing poly-nomials introduced by Mangontarum et al. [12] defined as e F m,a ( n ; x ) = n X k =0 k ! f W m,a ( n, k ) x k , (14)where f W m,a ( n, k ) are the noncentral Whitney numbers of the second kind with an exponentialgenerating function given by ∞ X n = k e F m,a ( n ; x ) z n n ! = me − az m − x ( e mz − . (15)Looking at (15), it is readily observed that e F m, ( n ; x ) = w n (cid:16) xm (cid:17) , e F m, − ( n ; x ) = F m, ( n ; x )and e F , − r ( n ; x ) = w n ( r, x ) . The numbers f W m,a ( n, k ) admit a variety of combinatorial properties which can be seen in[12]. These numbers appear to be a common generalization of (cid:8) nk (cid:9) and W m ( n, k ), as well asother notable numbers reported by the respective authors in [1, 5, 10, 11, 13]. It is importantto note that the noncentral Whitney numbers of the second kind is equivalent to the ( r, β )-Stirling numbers by Corcino [7] and the r -Whitney numbers of the second kind by Mez˝o[14].In the present paper, we establish some formulas for the noncentral Tanny-Dowling poly-nomials including sums of products and explicit formulas. These formulas are shown togeneralize the above-mentioned identities obtained by Kargın [9] for the geometric polyno-mials when the parameters are assigned with specific values. We also discuss some identitiesresulting from the said formulas. Now, the exponential generating function in (15) can be rewritten as ∞ X n =0 e F m,a ( n ; x ) z n n ! = 11 − xm ( e mz − · e − az . ∞ X n =0 e F m,a ( n ; x ) z n n ! = ∞ X n =0 m n w n (cid:16) xm (cid:17) z n n ! ∞ X n =0 ( − a ) n z n n != ∞ X n =0 n X k =0 (cid:18) nk (cid:19) w k (cid:16) xm (cid:17) m k ( − a ) n − k ! z n n ! . Comparing the coefficients of z n n ! yields the result in the next theorem. Theorem 1.
The noncentral Tanny-Dowling polynomials e F m,a ( n ; x ) satisfy the followingidentity: e F m,a ( n ; x ) = n X k =0 (cid:18) nk (cid:19) m k w k (cid:16) xm (cid:17) ( − a ) n − k . (16) Alternative proof of Theorem 1.
From [12, Theorem 10], the noncentral Whitney numbers ofthe second kind satisfy the following formula in terms of the Stirling numbers of the secondkind: f W m,a ( n, k ) = n X j =0 (cid:18) nj (cid:19) ( − a ) n − j m j − k (cid:26) jk (cid:27) . Multiplying both sides by k ! x k and summing over k gives the desired result.Before proceeding, we see that when m = 1 and a = − r , (16) becomes e F , − r ( n ; x ) = n X k =0 (cid:18) nk (cid:19) w k ( x ) r n − k := w n ( r ; x ) , which is precisely an identity obtained by Kargın [9, Equation (13)].By applying the exponential generating function in (15), ∞ X n =0 h e F m,a − m ( n ; x ) − e F m,a ( n ; x ) i z n n ! = me − ( a − m ) z m − x ( e mz − − me − az m − x ( e mz − mx (cid:20) me − az m − x ( e mz − − e − az (cid:21) = mx ∞ X n =0 e F m,a ( n ; x ) z n n ! − ∞ X n =0 ( − a ) n z n n != ∞ X n =0 mx (cid:16) e F m,a ( n ; x ) − ( − a ) n (cid:17) z n n ! . Comparing the coefficients of z n n ! gives e F m,a − m ( n ; x ) − e F m,a ( n ; x ) = mx h e F m,a ( n ; x ) − ( − a ) n i . The result in the next theorem follows by solving for x e F m,a − m ( n ; x ).4 heorem 2. The noncentral Tanny-Dowling polynomials e F m,a ( n ; x ) satisfy the followingrecurrence relation: x e F m,a − m ( n ; x ) = ( m + x ) e F m,a ( n ; x ) − ( − a ) n m. (17)Setting m = 1 and a = − r in (17) gives x e F , − r − ( n ; x ) = (1 + x ) e F , − r ( n ; x ) − r n which is exactly the following identity [9, Equation (14)]: xw n ( r + 1; x ) = (1 + x ) w n ( r ; x ) − r n . On the other hand, when a = 0 and a = m in (17), we get x e F m, − m ( n ; x ) = ( m + x ) w n (cid:16) xm (cid:17) (18)and ( m + x ) e F m,m ( n ; x ) = xw n (cid:16) xm (cid:17) − ( − m ) n +1 , (19)respectively. Applying (16) yields xm n n X k =0 (cid:18) nk (cid:19) w k (cid:16) xm (cid:17) = ( m + x ) w n (cid:16) xm (cid:17) (20)and ( m + x ) m n n X k =0 (cid:18) nk (cid:19) w k (cid:16) xm (cid:17) ( − n − k = xw n (cid:16) xm (cid:17) − ( − m ) n +1 . (21)These are generalizations of the results obtained by Dil and Kurt [8] using the Euler-Seidelmatrix method. That is, setting x = 1 and m = 1 gives n X k =0 (cid:18) nk (cid:19) w k = 2 w n and 2 n X k =0 (cid:18) nk (cid:19) ( − k w k = ( − n w n + 1 . The next theorem contains a formula for the sum of product of noncentral Tanny-Dowlingpolynomials for different values of a . Theorem 3.
The noncentral Tanny-Dowling polynomials satisfy the following relation: x n X k =0 (cid:18) nk (cid:19) e F m,a ( k ; x ) e F m,a ( n − k ; x ) = e F m, ¯ A ( n + 1; x ) + ¯ A e F m, ¯ A ( n ; x ) , (22) where ¯ A = a + a + m for real numbers a and a . roof. We start by taking the derivative of (15) with respect to z . That is, ∂∂z (cid:18) me − az m − x ( e mz − (cid:19) = me − az m − x ( e mz − · xme mz m − x ( e mz − − ame − az m − x ( e mz − . Replacing a with ¯ A = a + a + m yields ∂∂z me − ¯ Az m − x ( e mz − ! = ∞ X n = k e F m, ¯ A ( n + 1; x ) z n n !in the left-hand side while we get me − ¯ Az m − x ( e mz − · xme mz m − x ( e mz −
1) = me − a z m − x ( e mz − · me − a z m − x ( e mz − x ∞ X n = k n X k =0 (cid:18) nk (cid:19) e F m,a ( k ; x ) e F m,a ( n − k ; x ) z n n !and ¯ Ame − ¯ Az m − x ( e mz −
1) = ¯ A · ∞ X n = k e F m, ¯ A ( n ; x ) z n n !in the right-hand side. Combining the above equations and comparing the coefficients of z n n ! gives the desired result.When a = a = 0 in (22), x n X k =0 (cid:18) nk (cid:19) w k (cid:16) xm (cid:17) w n − k (cid:16) xm (cid:17) = e F m,m ( n + 1; x ) + m e F m,m ( n ; x ) . Applying (17) to the right-hand side of this equation gives x n X k =0 (cid:18) nk (cid:19) w k (cid:16) xm (cid:17) w n − k (cid:16) xm (cid:17) = xw n +1 (cid:0) xm (cid:1) − ( − m ) n +2 m + x + m xw n (cid:0) xm (cid:1) − ( − m ) n +1 m + x which can be simplified into the following identity:( m + x ) n X k =0 (cid:18) nk (cid:19) w k (cid:16) xm (cid:17) w n − k (cid:16) xm (cid:17) = w n +1 (cid:16) xm (cid:17) + mw n (cid:16) xm (cid:17) . (23)Obviously, this identity boils down to the result obtained by Kargın [9] in (9) when m = 1. Theorem 4.
For x = x , the following formula holds: n X k =0 (cid:18) nk (cid:19) e F m,a ( k ; x ) e F m,a ( n − k ; x ) = x e F m,a + a ( n ; x ) − x e F m,a + a ( n ; x ) x − x . (24)6 roof. Note that we can write me − a z m − x ( e mz − · me − a z m − x ( e mz −
1) = 1 x − x (cid:20) x me − ( a + a ) z m − x ( e mz − − x me − ( a + a ) z m − x ( e mz − (cid:21) . Following the same method used in the previous theorem leads us to the desired result.This theorem contains a formula for the sums of products of noncentral Tanny-Dowlingpolynomials for different values of x . When a = a = 0, (24) reduces to n X k =0 (cid:18) nk (cid:19) w k (cid:16) x m (cid:17) w n − k (cid:16) x m (cid:17) = x w n (cid:0) x m (cid:1) − x w n (cid:0) x m (cid:1) x − x . (25)It is clear to see that when m = 1, we recover the sum of products of geometric polynomialsin (10). In Theorem 1, we obtained an explicit formula that expresses the noncentral Tanny-Dowlingpolynomials in terms of the geometric polynomials. Now, with g n = a n e F m,a ( n ; x ) and f j = (cid:0) ma (cid:1) j w j (cid:0) xm (cid:1) , the binomial inversion formula f n = n X j =0 (cid:18) nj (cid:19) g j ⇐⇒ g n = n X j =0 ( − n − j (cid:18) nj (cid:19) f j (26)allows us to express the geometric polynomials w n (cid:0) xm (cid:1) in terms of the noncentral Tanny-Dowling polynomials as follows. w n (cid:16) xm (cid:17) = 1 m n n X j =0 (cid:18) nj (cid:19) a n − j e F m,a ( j ; x ) . (27)In this section, we will derive more explicit formulas for both polynomials.Using x − m in place of x in (15) gives ∞ X n = k e F m,a ( n ; x − m ) z n n ! = me − ( − a − m )( − z ) m + x ( e − mz − ∞ X n = k e F m, − a − m ( n ; − x ) ( − z ) n n ! . By comparing the coefficients of z n n ! , we get e F m,a ( n ; x − m ) = ( − n e F m, − a − m ( n ; − x ) . (28)7pplying (17) to the right-hand side gives e F m,a ( n ; x − m ) = ( − n " ( m − x ) e F m, − a ( n ; − x ) − a n m − x . Replacing − x and − a with x and a , respectively, and solving for e F m,a ( n ; x ) yields e F m,a ( n ; x ) = ( − n x e F m, − a ( n ; − x − m ) + ( − a ) n mm + x . By (14), we get the next theorem.
Theorem 5.
The noncentral Tanny-Dowling polynomials satisfy the following explicit for-mula: e F m,a ( n ; x ) = x n X k =0 ( − n + k k ! f W m, − a ( n, k )( m + x ) k − + ( − a ) n mm + x . (29)Setting a = 0 in f W m,a ( n, k ) allows us to express the noncentral Whitney numbers of thesecond kind in terms of (cid:8) nk (cid:9) . More precisely, when a = 0 in [12, Proposition 7], we can seethat f W m, ( n, k ) = m n − k (cid:26) nk (cid:27) . Thus, (29) becomes w n (cid:16) xm (cid:17) = x n X k =0 ( − n + k k ! m n − k (cid:26) nk (cid:27) ( m + x ) k − (30)when a = 0. Moreover, when m = 1, we recover the explicit formula in (11). The expression m n − k (cid:8) nk (cid:9) is actually called translated Whitney numbers of the second kind and is denotedby (cid:8) nk (cid:9) ( m ) . These numbers satisfy the recurrence relation given by [1, Theorem 8] (cid:26) nk (cid:27) ( m ) = (cid:26) n − k − (cid:27) ( m ) + mk (cid:26) n − k (cid:27) ( m ) and the explicit formula [13, Proposition 2] (cid:26) nk (cid:27) ( m ) = 1 m k k ! k X j =0 ( − k − j (cid:18) kj (cid:19) ( mj ) n . More properties of these numbers can be seen in [11]. With these, we may also write w n (cid:16) xm (cid:17) = x n X k =0 ( − n + k k ! (cid:26) nk (cid:27) ( m ) ( m + x ) k − , (31)an explicit formula for the geometric polynomials w n (cid:0) xm (cid:1) in terms of the translated Whitneynumbers of the second kind. 8ow, it can be shown that y − y (cid:18) e − a (2 z ) y − e mz + e − a (2 z ) y + e mz (cid:19) = e − a (2 z ) − (cid:16) y − (cid:17) ( e m (2 z ) − ) . Notice that the right-hand side is e − a (2 z ) − (cid:16) y − (cid:17) ( e m (2 z ) − ) = me − a (2 z ) m − (cid:16) my − (cid:17) ( e m (2 z ) − )= ∞ X n =0 n e F m,a (cid:18) n ; my − (cid:19) z n n ! . Also, in the left-hand side, we have e − a (2 z ) y − e mz = 1 y − ∞ X n =0 e F m, a (cid:18) n ; my − (cid:19) z n n !and e − a (2 z ) y + e mz = 1 y + 1 ∞ X n =0 e F m, a (cid:18) n ; − my + 1 (cid:19) z n n ! . Combining these equations and comparing the coefficients of z n n ! results to2 n +1 e F m,a (cid:18) n ; my − (cid:19) = y + 1 y e F m, a (cid:18) n ; my − (cid:19) + y − y e F m, a (cid:18) n ; − my + 1 (cid:19) . Note that if we set x = my − , then y = m + xx . Hence, skipping the tedious computations allowus to write( m + 2 x ) e F m, a ( n ; x ) = 2 n +1 ( m + x ) e F m,a (cid:18) n ; x m + 2 x (cid:19) − m e F m, a (cid:18) n ; − mxm + 2 x (cid:19) . The next theorem is obtained by applying (14).
Theorem 6.
The noncentral Tanny-Dowling polynomials satisfy the following explicit for-mula: e F m, a ( n ; x ) = n X k =0 k ! x k " n +1 ( m + x ) x k f W m,a ( n, k ) + ( − m ) k +1 f W m, a ( n, k )( m + 2 x ) k +1 . (32)Since it is already known that f W m, ( n, k ) = (cid:8) nk (cid:9) ( m ) , then when a = 0, the right-hand sidecan be expressed in terms of the translated Whitney numbers of the second kind. That is, w n (cid:16) xm (cid:17) = n X k =0 k ! x k (cid:26) nk (cid:27) ( m ) (cid:20) n +1 ( m + x ) x k + ( − m ) k +1 ( m + 2 x ) k +1 (cid:21) . (33)9astly, when m = 1, we recover the explicit formula in (12).Finally, we will end by mentioning an explicit formula for e F m,a ( n ; x ) established in [12,Theorem 19] that is given by e F m,a ( n ; x ) = mm + x ∞ X k =0 (cid:18) xm + x (cid:19) k ( mk − a ) n . (34)This explicit formula entails interesting particular cases. For instance, when a = 0, w n (cid:16) xm (cid:17) = m n +1 m + x ∞ X k =0 (cid:18) xm + x (cid:19) k k n . (35)When m = 1 and then x = 1, we get formulas for the ordinary geometric polynomials andnumbers. That is, w n ( x ) = 1 x + 1 ∞ X k =0 (cid:18) xx + 1 (cid:19) k k n (36)and w n = ∞ X k =0 k n k +1 . (37) References [1] H. Belbachir and I. Bousbaa, Translated Whitney and r -Whitney numbers: a combina-torial approach, J. Integer Seq. (2013), Article 13.8.6.[2] M. Benoumhani, On Whitney numbers of Dowling lattices, Discrete Math. (1996),13–33.[3] M. Benoumhani, On some numbers related to Whitney numbers of Dowling lattices,
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